To see if a number, say $562437487$, is divisible by $3$, you just add up the digits of its decimal representation, and see if the result is divisible by $3$ $(5+6+2+4+3+7+4+8+7 = 46$, so it is not divisible by $3$). To see if the same number is divisible by $11$, you can do this: subdivide the number into pairs of digits, from the right hand end $(87, 74, 43, 62, 5)$, add these numbers, and see if the sum is divisible by $11$ (if it is too big, repeat). How about $37$? To see if the number is divisible by $37$, subdivide it into triples from the end $(487, 437, 562)$, add these up, and see if the sum is divisible by $37$. This is true for any prime $p$ other than $2$ and $5$. That is, for any prime $p\not =2,5$ there is an integer $r$ such that in order to see if $p$ divides a decimal number $n$, we break $n$ into $r−tuples$ of decimal digits (starting from the right-hand end), add up these $r−tuples$, and check if the sum is divisible by $p$.
(a) What is the smallest such $r$ for $p = 13$?
(b) What is the smallest such $r$ for $p = 17$?
(c) Prove that $r$ is a divisor of $p−1$.
My understanding: Somehow I have understood I have to prove that $p$ divides $10^r - 1$. Using which part $(a)$ and $(b)$ can easily be solved. But , I am not able to mathematically prove this result that $p$ divides $10^r - 1$.
